Three-loop vacuum integral with four-propagators using hypergeometry *

A hypergeometric function is proposed to calculate the scalar integrals of Feynman diagrams. In this study, we verify the equivalence between the Feynman parametrization and the hypergeometric technique for the scalar integral of the three-loop vacuum diagram with four propagators. The result can be described in terms of generalized hypergeometric functions of triple variables. Based on the triple hypergeometric functions, we establish the systems of homogeneous linear partial differential equations (PDEs) satisfied by the scalar integral of three-loop vacuum diagram with four propagators. The continuation of the scalar integral from its convergent regions to entire kinematic domains can be achieved numerically through homogeneous linear PDEs by applying the element method.